For that matter, I don't think we ever saw ships in the TOS movies engaged in combat at warp, so we don't know that it was possible with the technology they were using at that time. We assume it was because that's what we're used to, but we have no proof.

I was about to declare that there is warp-speed combat in "Errand of Mercy," but rechecking the transcript, the Enterprise is attacked by the Klingons after stopping to decode a Starfleet message, before setting course for Organia. Almost as if the Klingons were waiting for them to drop out of warp!